MEDB 5502, Module 13, Bayesian Statistics

Topics to be covered

  • What you will learn
    • Classical statistics
    • Bayesian statistics
    • Prior distribution
    • Likelihood
    • Posterior distribution
    • Applications of Bayesian statistics

Classical statistics, 1 of 3

  • \(H_0:\ \pi_1=\pi_2\)
  • \(H_1:\ \pi_1 \ne \pi_2\)
    • \(\pi_1\), \(pi_2\) are fixed constants

Classical statistics, 2 of 3

  • \(T=\frac{\hat p_1-\hat p_2}{s.e.}\)
  • p-value = \(P[Z > T]\)
    • probability of sample results or more extreme
    • p-value is not \(P[H_0]\)

Classical statistics, 3 of 3

  • 95% confidence interval
    • \(\hat p_1-\hat p_2 \pm Z_{\alpha/2}s.e.\)
    • Range of plausible values
  • Probability statements not possible.

Break #1

  • What you have learned
    • Classical statistics
  • What’s coming next
    • Bayesian statistics

A simple example of Bayesian data analysis.

  • ECMO study
  • Treatment versus control, mortality endpoint
    • Treatment: 28 of 29 babies survived
    • Control: 6 of 10 babies survived
    • Source: Jim Albert in the Journal of Statistics Education (1995, vol. 3 no. 3).

Wikipedia introduction

  • P(H|E) = P(E|H) P(H) / P(E)
    • H = hypothesis
    • E = evidence
    • P(H) = prior
    • P(E|H) = likelihood
    • P(H|E) = posterior

Prior distribution

  • Degree of belief
    • Based on previous studies
    • Subjective opinion (!?!)
  • Examples of subjective opinions
    • Simpler is better
    • Be cautious about subgroup analysis
    • Biological mechanism adds evidence
  • Flat or non-informative prior

Break #2

  • What you have learned
    • Bayesian statistics
  • What’s coming next
    • Prior distribution

Lay out the parameters

Half of probability on the diagonal

  • P(H|E) = P(E|H) P(H) / P(E)

Half of probability off the diagonal

  • P(H|E) = P(E|H) P(H) / P(E)

Difuse prior

  • P(H|E) = P(E|H) P(H) / P(E)

Break #3

  • What you have learned
    • Prior distribution
  • What’s coming next
    • Likelihood

Likelihood

  • P(H|E) = P(E|H) P(H) / P(E)

Break #4

  • What you have learned
    • Likelihood
  • What’s coming next
    • Posterior distribution

Multiply

  • P(H|E) = P(E|H) P(H) / P(E)

Standardize

  • P(H|E) = P(E|H) P(H) / P(E)

Main diagonal of posterior probabilities

  • P(H|E) = P(E|H) P(H) / P(E)

Upper triangle of posterior probabilities

  • P(H|E) = P(E|H) P(H) / P(E)

Break #5

  • What you have learned
    • Posterior distribution
  • What’s coming next
    • Applications of Bayesian statistics

Applications of Bayesian statistics

  • Incorporate previous research
    • Controls from earlier studies
  • Random coefficient models
  • Missing data
  • Non-standard measures
    • Ranking the best

Summary

  • What you have learned
    • Classical statistics
    • Bayesian statistics
    • Prior distribution
    • Likelihood
    • Posterior distribution
    • Posterior distribution
    • Applications of Bayesian statistics